Quick estimate of CSR-induced microbunching instability in a multibend transport beamline

Microbunching instability (MBI) driven by short-range wakefields in high-brightness electron beams has been an active area of research over the past decade. While most existing studies focus on single-pass or linear accelerators --- particularly few-dipole bunch compressor chicanes --- MBI studies in multi-bend transport lines has relied predominantly on time-consuming numerical simulations. In this work, we present a quick estimate for evaluating MBI gain in generic multi-bend beamlines, thereby avoiding computational costs. Starting from Volterra integral equation governing the bunching factor, we first find the optimal wavelength and introduce physically motivated simplifications to derive the maximum gain. A gain spectrum is then constructed based on physical insights into MBI amplification mechanisms. The results show good agreement with detailed numerical calculations from Vlasov solver. The developed approach enables quick and reasonably accurate estimates of the MBI gain using only the lattice optics functions and the initial beam parameters, offering a practical tool for beamline design and mitigation of MBI.